Problem: Solve for $x$ : $ 6|x - 10| - 6 = 3|x - 10| + 6 $
Answer: Subtract $ {3|x - 10|} $ from both sides: $ \begin{eqnarray} 6|x - 10| - 6 &=& 3|x - 10| + 6 \\ \\ { - 3|x - 10|} && { - 3|x - 10|} \\ \\ 3|x - 10| - 6 &=& 6 \end{eqnarray} $ Add ${6}$ to both sides: $ \begin{eqnarray} 3|x - 10| - 6 &=& 6 \\ \\ { + 6} &=& { + 6} \\ \\ 3|x - 10| &=& 12 \end{eqnarray} $ Divide both sides by ${3}$ $ \dfrac{3|x - 10|} {{3}} = \dfrac{12} {{3}} $ Simplify: $ |x - 10| = 4$ Because the absolute value of an expression is its distance from zero, it has two solutions, one negative and one positive: $ x - 10 = -4 $ or $ x - 10 = 4 $ Solve for the solution where $x - 10$ is negative: $ x - 10 = -4 $ Add ${10}$ to both sides: $ \begin{eqnarray} x - 10 &=& -4 \\ \\ {+ 10} && {+ 10} \\ \\ x &=& -4 + 10 \end{eqnarray} $ $ x = 6 $ Then calculate the solution where $x - 10$ is positive: $ x - 10 = 4 $ Add ${10}$ to both sides: $ \begin{eqnarray} x - 10 &=& 4 \\ \\ {+ 10} && {+ 10} \\ \\ x &=& 4 + 10 \end{eqnarray} $ $ x = 14 $ Thus, the correct answer is $x = 6 $ or $x = 14 $.